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Chapter9

AsynchronousSequentialLogic

9.1INTRODUCTION

A sequentialcircuithspecifiedbya timesequence ofinputs.outputs.andinternalstates.In synchronoussequentialcircuits.the changeofinternal state occursin responsetothe syn chronizedclockpulses.Asynchronoussequentialcircuitsdo notuseclockpulses.Thechange ofinternalstateoccurswhenthereisa changein theinput variables.Thememoryelementsin synchronoussequentialcircuitsare clockedflip-flops.

Thememoryelementsin asynchronous

sequentialcircuitsareeitherunclockedflip-flops or time-delayelements.

Thememory cepe

biliryofatime-delaydevicedependson the finite amount of timeitlakesfor thesignal10 combinationalcircuitwithfeedback. Thedesignofasynchronoussequentialcircuits ismoredifficult than thatofsynchronouscir cuitsbecause ofthetimingproblemsinvolved in thefeedbackpath.In a properlydesigned synchronoussystem.timingproblemsare eliminatedbytriggeringallflip-flops with thepulse edge. Thechange fromone state to the nextoccursduringthe shorttimeofthepulse transi tion.Sincetheasynchronouscircuitdoesnot use aclock.the state ofthesystemis allowedto chang e immediatelyaftertheinputchanges.Care must betakento ensure thateachnewstate keepsthecircuitina stableconditioneventhougha feedback pathexists. Asynchronoussequential circuitsareuseful in avariety ofapplications. Theyareused when speed ofoperationis important.especiallyinthosecaseswhere thedigital system must reo spendquickly without havingtowait for aclockpulse.They aremoreeconomicaltouse in smallindependentsystemsthatrequireonlya few components,as it maynotbepracticalto go to the expense ofproviding a circuitforgeneratingclockpulses.Asynchronous circuitsare usefulinapplications where theinputsignalsto the systemmay changeat any time. inde pendently ofan internalclock.Thecommunicationbetweentwounits.eachhavingitsown 415

416Chapter 9AsynchronousSequentialLogic

independentclock, mustbedone withasynchronouscircuits.Digitaldesignersoften produce amixedsystemin which some panofthe synchronoussystemhasthecharacteristicsofan asyn chronous circuit Knowl edgeofasynchronoussequentiallogic behaviorishelpfulinverifying that the totaldigitalsystemisoperating in thepropermanner.

Figure9.1showstheblockdiagram

acombinationalcircuitanddelayelementsconnectedto form feedback loops.There aren inputvariables.moutput variables.andk:internalstates.The delayelementscanbevisualized asproviding short-termmemory forthesequentialcircuit.In agate-type circuit,the propaga tiondelaythat exists inthe combinationalcircuitpath from inputto outputprovidessufficient delayalongthefeedbackloopso thatnospecific delayelements areactuallyinsertedintothc feedbackpath.Thepresent-stateandnext-statevariablesinasynchronous sequentialcircuits arccustomarily called lionvariables should not beconfusedwith the excitabletableusedinthe designofclockedse quentialcircuits. einputmouipulvariablesvariables y,Y, ksecondarykexcitation variables y,Y, variables (present(neX!slale) stale) y.Y. Delay

FI(i,URE9.1

Blockdiagramof anasynchronoussequentialcircuit

Section9.2AnalysisProcedure417

When aninput variable changes invalue,theysecondary variablesdonot change instan taneousty. Itlakesa certainamountoftime forthe signal10 propagatefromthe inputtermi nals,through thecombinationalcircuit.totheYexcitation variables.whichgenerate newvalues for thenext state.Thesevaluespropagatethroughthe delayelementsandbecome the new presentstatefor the secondaryvariables. Note the distinctionbetweenthe v'sand theY's.In the steady-state condition, theyare thesame, butduringtransitiontheyare not. For a givenvalue of inputvariables.thesystem is stableif the circuitreaches a steady-stateconditionwith Yi=Y; fori=1.2,.. .•k.Otherwise.thecircuitisinaconunuoustransitionandis saidtobeunsta ble. Itisimportant10realizethai atransitionfromonestablestateto anotheroccursonlyin re spousetoachange inaninp utvariable.Thisis incontrast tosynchronoussystems. inwhich slatetransitionsoccurinresponsetothe applicationofaclockpulse. Toensureproper operation,asynchronous sequentialcircuitsmust beallowed toattaina sta blestate beforethe input ischangedtoanewvalue. Becauseof delaysin the wiresand the gate circuits,it isimpossibleto havetwoor more input variableschange atexactly the sameinstant of time withoutanuncertaintyastowhich one changes first.Therefore,simultaneous changes of two ormorevariables areusuallyprohibited. Thisrestrictionmeansthatonlyone input variable can changeata nyone time andthetimebetweentwoinputchangesmustbe longerthanthetime itlakesthecircuit toreachastablestate.Such oper ation.defined esfundamentalmode.assumes thaitheinputsignals changeoneat atimeandonlywhen the circuit isinastablecondition.

9.2ANALYSISPROCEDURE

The analysisofasynchronoussequentialcircuits consistsofobtaininga tableor adiagramthat describesthe sequence of internal statesand outputsasa functionof changesin the Inputvari ables.Alogicdiagrammanifeststhebehaviorofan asynchronoussequentialcircuitif ithasone or morefeedbackloopsorifit includesunclocked flip-flops.Inthis section.we will investi gate thebehaviorofasynchronous sequential circuitsthathavefeedbackpathswithoutem ployingflip-flops.UnclockedFlip-flops arecalled latches,andtheir usein asynchronous sequential circuitswillbe explained in thenext section. The analysisprocedurewill be presentedby meansof three specific examples.The firstex ample introduces thetransitiontable,the second definesthe flow table, and the thirdinvesti gatesthestabilityofasynchronous sequentialcircuits.

TransitionTable

An exampleof anasynchronous sequentialcircuit withonlygates isshowninFig.9.2.Thedi agramclearly shows two feedbackloopsfromthe ORgate outputsback10the ANDgate in puts.Thecircuitconsists of oneinput variable xandtwointernalstates.The internalstates havetwo excitation variables, Y,andY2.and twosecondaryvariables, Yl and)'2.Thedelay as sociated witheach feedbackloopis obtained fromthepropagationdelaybetweeneach

Yinput

and itscorresponding youtput.Eachlogic gateinthe path introducesapropagationdelayof about2toIOns.Thewiresthatconductelectrical signals introduce approximatelyaI-nsdelay foreachfoot ofwire.Thus,no additionalexternal delayelementsarenecessary whenthe com binationalcircuitand the wiresinthefeedbackpath provide sufficientdelay.

418 Chapter 9AsynchronousSequentiallogic

Y, Y,

FIGURE9.2

Exampleof an asynchronoussequential circuit

The analysisofthecircuitstarts witha considerationof theexcitationvariables as outputs and the secondaryvariables asinputs.Wethen derive the Booleanexpressions for theexcita tion variablesasa functionof theinputandsecondary variables.These expressions.readily ob tained fromthelogicdiagram.are Y l =XYI+x'n

Y2=xYI+.t'n

Thenext stepistoplottheY

1 andY2functions inamap. as shownin Fig.9.3(a)and(b).The encoded binary valuesof the)'variables areused forlabeling the rows.andthe input .rvari ableisused todesignatethecolumns.This configurationresults inaslightly differentthree variablemap fromthe one usedin previous chapters. However,itis still a validmap.andsuch ,y, 0 1 00@01 0111
e 11 e 10 10 00 ,y, 01 000 1 0111
1110

10 0 0

yy 0 1 000 0 01 10 1111
1001
y (a) Mapfor

Yl-XYI+X'Y2

(b)Map for(e)Transitiontable

Y2=xy'1+X'y:

fiGURE 9.3

Mapsandtransitiontableforthecircuit of Fig.9.2

Section9.2AnalysisProcedure419

unlikewhatwasdoneinpreviouschapters.the variablesbelongingtotheappropriate squares are notmarkedalongthe sides ofthe map. Tbetransitiontablesbcwnin Fig.9.3(c)isobtainedfrom the mapsbycombiningthe binary values incorresponding squares,Thetransitiontable shows the value of

Y=flY.!insideeach

square .Thefin!bitofYisobtainedfrom thevalueoffl.and thesecond bit is obtained from thevalueof Ylin the same squareposition.For a staretobestable, thesecondaryvariablesmust match theexcitationvariables(i.e.•the valueofYmustbethe sameasthatofY=YI)',2)'Those entriesin the transitiontablewhere Y=Yare circled to indicateastablecondition.An uncir cledentryrepresentsan unstablestate.

Now considertheeffect

ofachangein the input variable.The squarefor.r=0 andy=00 in the transitiontableshowsthat f=00.Sincefrepresentsthenext valueofy.thisisasta blecondition, If .rchangesfrom0 toIwhiley=00.the circuitchangesthevalueofYto01. This representsa temporaryunstablecondition,becauseYisnot equal10the presentvalueof j-,Whathappensnextisthatas soonasthesignal propagatesto makeY=01,the feedback pathin the circuitcauses achange in yto 01. This change ismanifestedin the transitiontable bya transitionfromthe first row (y=00)to thesecondrow,wherey=01.Nowthaty=Y, thecircuit reaches astable conditionwithan input of.r=I.In general.ifa change inthe input takes the circuitto an unstableSlate. the valueof ywillchange(whilethat of.rremain.. thesame)until itreaches a Mable(circled)state. Usingthistypeofanalysisfortheremaining squares ofthetransitiontable.wefindthatthecircuit repeatsthe sequenceofstates 00,01.II.

10when the input repeatedlyalternatesbetween0 and

I. Note the difference betweena synchrorccsand anasynchronous sequential circuit.Ina syn chronoussystem. mepresentstate istotallyspecified bytheflip-flop valuesanddues notchange if theinputchanges while the clockpulseis inactive, Inan asynchronouscircuit.the internal state can change immediately aftera change intheinput,Becauseofthis rapidchange,ilissome timesconvenient tocombine theinternalstate....iththeinputvaluetogetherandcallit the10101 stateofthecircuit.Thecircuit whose transition tableis showninFig.9.3(c)hasfourstabletotal =000.011.110.and101-andfour unstable totalstates-OOI.OIO. III.and100. Thetransitiontableofasynchronous sequentialcircuit..is similartothe state tableusedfor synchronouscircuits.Ifwe regardthe secondary variablesasthepresentstate andtheexci tationvariablesasthenextstate.weobtain thestatetablesho wn in Table 9.1.Thistablepro videsthesameinformationasthe transitiontable.There isonerestrictionthat appliesto the

Table 9.1

State TablefortheCircuitofFig,9.1

NeatState

Present

StateK·0K=1

0 00001

0II I0I

I0 0 0I0

I I II I0

420 Chapter 9 AsynchronousSequentiallogic

asynchronous case.butnot thesynchronouscase:In theasynchronoustransitiontable.there usually isatleastonenext-stateentrythatisthe sameasthe present-statevalue ineach row. Otherwise.alithe totalstatesin thatrow willbe unstable. Theprocedure forobtaininga transitiontablefromthecircuitdiagramofanasynchronous sequentialcircuit isasfollows:

1.Determine allfeedback loopsin thecircuit.

2. Designate the output of each feedback loopwithvariable

Y;and its correspondinginput

with jyfori=1.2•...•k,wherekisthenumber offeedbackloopsin the circuit.

3.Derive the Boolean functionsof all Y's asa functionof theexternalinputsandthe

.'"'!C>.

4.PloteachYfunction inamap.usingtheyvariablesfor the rowsandthe external inputs

forthecolumns. S.Combine allthemapsintoone tableshowingthevalueof Y=Y 1 Y 2 •••Y i insideeach square.

6. Circlethose valuesof

Yineach square that areequaltothe valueof.'"=inthe samerow. Once the transition table isavailable. the behavior ofthecircuitcan be analyzed byobserving thestaletransitionas afunctionofchangesin the inputvariables.

FlowTable

Duringthedesignofasynchronoussequentialcircuits.itismore convenient tonamethestates bylettersymbolswithout makingspecific reference totheirbinaryvalues.Such a tableis called aflowtableand is similar toa transitiontable.except thattheinternalstatesaresym bolizedwith lettersratherthan binary numbers.Theflow tablealsoincludestheoutput values of the circuitfor eachstablestate. Examples offlowtablesareshown in Fig. 9.4.The onein Fig. 9.4(a)has fourstates.des ignatedbytheletters a.b.c.andd.Itreducestothetransition tableofFig.9.3(c)ifweassign x y 0I 0 b b c 0 0 d d aquotesdbs_dbs17.pdfusesText_23

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